1. Field of the Invention
The computations of the physical properties of objects such as how they reflect and/or transmit energy and other physical effects require computer memory and processing time. For many problems, especially for large or complicated situations, these computations take significant computer resources. Solving large problems using existing methods may not even be possible on available computers. A new method is described for creating overlapping functions that have specified properties. These overlapping functions may be used to make computations or simulations of physical and other effects more efficient. For example, it may be desired to create orthogonal or nearly orthogonal functions where many of them radiate weakly (or strongly) to some or all regions of space. Such functions may sometimes be known a priori for a specific regular ordered situations along with a specific property. However, such functions are not known for other properties to be achieved or for other geometries, such as for irregular curved surfaces with an irregular boundary. This invention discloses numerical methods for creating such functions for various properties and for various geometries and discloses methods for their use in the design and/or construction of structures with desired properties. For example, these overlapping functions may allow the prediction of physical properties such as mechanical stiffness, acoustic scattering or electromagnetic scattering. These properties may be predicted for a proposed design, that design may be modified based on this prediction, and an improved physical device may then be constructed.
2. Description of the Related Art
Overlapping functions can be very useful when they are available. For example, a time series is sometimes examined by taking successive Fourier transforms of successive pieces of that time series. Often, the time series is partitioned into sub series, and a discrete Fourier transform is taken of each sub series. Here the word partitioned implies that each sub series does not overlap any other sub series. It is known that when one takes a discrete Fourier transform of non-periodic data with a finite length, the Gibbs Phenomena makes that series converge slowly. This phenomenon is a result of the fact that any finite Fourier series or transform (for discrete or continuous data respectively) is smooth and periodic while a sample of a time series, when considered as a periodic function, is discontinuous at it edges.
In the late 1980s, a method was discovered for generating overlapping orthogonal functions that produce a smoother version of the partitioned Fourier transform. In the context of the history of mathematics, this is a very recent result. These Malvar wavelets, or local cosines as they are also called, were developed for one dimensional functions such as time series. The use of a tensor product allows this result to be generalized to some highly regular two dimensional (and higher dimensional) surfaces. For example, this generalizes it to a planar surface for certain partitionings of that surface into regular rectangular regions.
This invention discloses a method that applies to irregular regions and to a variety of desired properties. It also allows the optimization (or the production of a significant amount) of a user specified property. This property may be chosen to be the same or different from the property that Malvar Wavelets optimize. Often, by allowing overlap between the functions of one region with another (or others), this optimization (or increase) can be achieved to a larger degree.
This invention also discloses numerical methods for culling out extra functions that (exactly or approximately) duplicate others. For example, consider two regions that neighbor each other. Here a region might be one part of a partition (it might also have a more general meaning). There might be one or more functions associated with each region, and each function might overlap into another possibly neighboring region or regions. Since one function associated with each region actually exist partly in say both regions, they might even be the same function. Similarly, if there are a number of functions associated with each region, the space generated by (linear combinations of) the functions in one region might either exactly or approximately overlap with the space generated by the neighboring region. This invention provides methods for minimizing (or eliminating) these extra functions, and for doing so in a way that optimizes (or tends to maintain) a desired property (or the desired properties). In some cases (but not in others) this represents finding an orthogonal (or nearly orthogonal) set of functions. In some cases, these functions are identified with regions as described earlier in this paragraph. In other cases, there is no clear such identification.
One example of a desired property concerns features of the radiation from electric currents flowing on a surface (or within a volume). For example, modern aircraft sometimes use their skin (or outer surface) as an antenna. The shape of that surface might be determined primarily by airflow considerations, rather than by antenna considerations. Nevertheless, such a complicated (or at least curved) surface might then be used as an antenna.
It is well known from the design of standard antennas that one can produce a beam with smaller sidelobes when the electric current decreases in magnitude towards the edges of that antenna. The sidelobes are analogous to the peaks of the Gibbs phenomena, and often represent relatively strong radiation in an undesired direction. Antenna design textbooks contain a large number of “taper functions” and describe the how these tapers may be used to reduce the strength of the sidelobes. For standard antennas, it is generally desired to keep these sidelobes small, so that nearly all of the power is contained within the “main beam”. Prior art has used several known taper functions on individual antennas.
The generation of overlapping excitations that produce specified properties such as narrow beams has many uses. For example, a large antenna might be thought of as a group of smaller antennas. One might design a large antenna by thinking of it as a collection of smaller antennas. If each produces a narrow beam, then one might ask how those narrow beams might be combined to produce an even narrower beam. This could be used to design antennas. These same principles also can be used in numerical computer programs that calculate electromagnetic effects. In such calculations the interactions between parts of a physical body may be computed using the analogy of those parts to antennas. These principles also apply to sound scattering and propagation, and to other effects. For example, they apply to the transmission of any form of energy, to physical effects, and to information flow. They apply to directional phenomena where the concept of a direction also involves a distance, such as a nearby region verses a distant region, and to properties other than directional characteristics. This invention relates, among other things, to new methods for producing overlapping currents or sources, where the currents for one antenna physically overlap the currents for another small antenna. It also relates to optimizing other properties using overlapping functions.
The use of a collection of (sub) antennas to design a larger antenna may provide efficiencies over numerical methods that have more parameters, such as the use of all of the degrees of freedom of each (sub) antenna. Methods that use high frequency approximations also reduce the number of degrees of freedom. Also, the use of a collection of sub antennas may allow a designer to have insight, or it may be used to synthesize a better performing antenna. Thus, this may not only allow a more efficient computation of an antenna's properties, but it may also aid in the design (synthesis) of more efficient antennas.
Overlapping functions may be created, for example, along a one dimensional surface. This invention provides a general method for constructing these and related functions. This method applies to general geometries such as creating overlapping functions defined on multidimensional surfaces and/or throughout multidimensional volumes. It also applies to general properties that may be optimized or made relatively strong.